Friday, April 27, 2012

On Equilibrium, Disequilibrium, and Rational Expectations

There's been some animated discussion recently on equilibrium analysis in economics, starting with a provocative post by Noah Smith, vigorousresponses by Roger Farmer and JW Mason, and some very lively comment threads (see especially the smart and accurate points made by Keshav on the latter posts). This is a topic of particularinterest to me, and the debate gives me a welcome opportunity to resume blogging after an unusually lengthy pause.

As Farmer's post makes clear, equilibrium in an intertemporal model requires not only that individuals make plans that are optimal conditional on their beliefs about the future, but also that these plans are mutually consistent. The subjective probability distributions on the basis of which individuals make decisions are presumed to coincide with the objective distribution to which these decisions collectively give rise. This assumption is somewhat obscured by the representative agent construct, which gives macroeconomics the appearance of a decision-theoretic exercise. But the assumption is there nonetheless, hidden in plain sight as it were. Large scale asset revaluations and financial crises, from this perspective, arise only in response to exogenous shocks and not because many individuals come to realize that they have made plans that cannot possibly all be implemented.

Farmer points out, quite correctly, that rational expectations models with multiple equilibrium paths are capable of explaining a much broader range of phenomena than those possessed of a unique equilibrium. His own work demonstrates the truth of this claim: he has managed to develop models of crisis and depression without deviating from the methodology of rational expectations. The equilibrium approach, used flexibly with allowances for indeterminacy of equilibrium paths, is more versatile than many critics imagine.

Nevertheless, there are many routine economic transactions that cannot be reconciled with the hypothesis that individual plans are mutually consistent. For instance, it is commonly argued that hedging by one party usually requires speculation by another, since mutually offsetting exposures are rare. But speculation by one party does not require hedging by another, and an enormous amount of trading activity in markets for currencies, commodities, stock options and credit derivatives involves speculation by both parties to each contract. The same applies on a smaller scale to positions taken in prediction markets such as Intrade. In such transactions, both parties are trading based on a price view, and these views are inconsistent by definition. If one party is buying low planning to sell high, their counterparty is doing just the opposite. At most one of the parties can have subjective beliefs that are consistent with with the objective probability distribution to which their actions (combined with the actions of others) gives rise.

If it were not for fundamental belief heterogeneity of this kind, there could be no speculation. This is a consequence of Aumann's agreement theorem, which states that while individuals with different information can disagree, they cannot agree to disagree as long as their beliefs are derived from a common prior. That is, they cannot persist in disagreeing if their posterior beliefs are themselves common knowledge. The intuition for this is quite straightforward: your willingness to trade with me at current prices reveals that you have different information, which should cause me to revise my beliefs and alter my price view, and should cause you to do the same. Our willingness to transact with each other causes us both to shrink from the transaction if our beliefs are derived from a common prior.

Hence accounting for speculation requires that one depart, at a minimum, from the common prior assumption. But allowing for heterogeneous priors immediately implies mutual inconsistency of individual plans, and there can be no identification of subjective with objective probability distributions.

The development of models that allow for departures from equilibrium expectations is now an active area of research. A conference at Columbia last year (with Farmer in attendance) was devoted entirely to this issue, and Mike Woodford's reply to John Kay on the INET blog is quite explicit about the need for movement in this direction:

The macroeconomics of the future... will have to go beyond conventional late-twentieth-century methodology... by making the formation and revision of expectations an object of analysis in its own right, rather than treating this as something that should already be uniquely determined once the other elements of an economic model (specifications of preferences, technology, market structure, and government policies) have been settled.

There is a growing literature on heterogenous priors that I think could serve as a starting point for the development of such an alternative. However, it is not enough to simply allow for belief heterogeneity; one must also confront the question of how the distribution of (mutually inconsistent) beliefs changes over time. To a first approximation, I would argue that the belief distribution evolves based on differential profitability: successful beliefs proliferate, regardless of whether those holding them were broadly correct or just extremely fortunate. This has to be combined with the possibility that some individuals will invest considerable time and effort and bear significant risk to profit from large mismatches between the existing belief distribution and the objective distribution to which it gives rise. Such contrarian actions may be spectacular successes or miserable failures, but must be accounted for in any theory of expectations that is rich enough to be worthy of the name.

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Some of the issues discussed here are explored at greater length in an essay on market ecology that I presented at a symposium in honor of Duncan Foley last week. Duncan was among the first to see that the rational expectations hypothesis implicitly entailed the assumption of complete futures markets, and would therefore be difficult to "reconcile with the recurring phenomena of financial crisis and asset revaluation that play so large a role in actual capitalist economic life."

True heterogeneity of priors in any substantive sense allowing for differences of understanding, judgment and changes in understand including true open ended learning RULES OUT tractability as a necessity of logic.

Formal systems and formal math rule out he very thing that is of substantive interest.

Faking "science" via the tractability imperative can't get you around this. In fact, it's a lead wall blocking scientific understanding and advance.

You have yet to explain what is the point or purpose of constructing such a "model" -- exactly the problem of the original equilibrium "modeling" imperative. What is it for, what is its use? How does it help us in identifying a problem raising pattern in our experience, or how does it help us provide a sound contingent causal explanation for that pattern?

Economists FORGOT to address this central and original question.

They had too much fun and too much professional reward just building the constructs -- to what purpose, Who Cares ...

So what is the purpose of doing this? What role does any of this play in the identification of a problem raising pattern in our experience, or in providing a contingent causal explanation accounting for that pattern.

Economists at least need to HINT at an answer:

"it is not enough to simply allow for belief heterogeneity; one must also confront the question of how the distribution of (mutually inconsistent) beliefs changes over time. To a first approximation, I would argue that the belief distribution evolves based on differential profitability: successful beliefs proliferate, regardless of whether those holding them were broadly correct or just extremely fortunate. This has to be combined with the possibility that some individuals will invest considerable time and effort and bear significant risk to profit from large mismatches between the existing belief distribution and the objective distribution to which it gives rise."

"Farmer points out, quite correctly, that rational expectations models with multiple equilibrium paths are capable of explaining a much broader range of phenomena than those possessed of a unique equilibrium."

I haven't read the threads leading up to this, so my comment may well be off:

On a purely theoretical level, a model with a continuum of equilibria, say, can explain not just a broader range of phenomena, but any kind of phenomena. But that could be a weakness rather than a strength.

For multiple equilibria models to be superior, you need overwhelming evidence of multiple equilibria in the real world. On this, there was a debate between Krugman and Obstfeld, Calvo and others, about the empirical relevance of multiple equilibria models in models of balance of payments crises, and Krugman conceded eventually that Obstfeld and Calvo, etc. had won the arguments, but that took an awful lot of empirical evidence.

I'm not sure that the likes of Farmer have much to show in the way of empirical evidence. At least from what I recall seeing in his little MIT press book on OLG models and fiat money and so on, that is a while back already.

Finding clear evidence for multiple equilibria is intrinsically hard - if one of these is selected in the real world, how can one really know if others exist? Estimation can only be done locally, around our current state. But sometimes the logic of multiplicity is compelling. For example, a large enough increase in interest rates virtually guarantees default by a debtor, and this in turn justifies the high rates. Not sure how much evidence there is for the Farmer models though.